Universitext - The Eye of Constructive... · Lume ce gˆandeaˆın basme ¸si vorbeaˆın poezii,...

Universitext

Editorial Board(North America):

S. Axler

K.A. Ribet

Universitext

Editors (North America): S. Axler and K.A. Ribet

Aguilar/Gitler/Prieto: Algebraic Topology from a Homotopical ViewpointAksoy/Khamsi: Nonstandard Methods in Fixed Point TheoryAndersson: Topics in Complex AnalysisAupetit: A Primer on Spectral TheoryBachman/Narici/Beckenstein: Fourier and Wavelet AnalysisBadescu: Algebraic SurfacesBalakrishnan/Ranganathan: A Textbook of Graph TheoryBalser: Formal Power Series and Linear Systems of Meromorphic OrdinaryDifferential EquationsBapat: Linear Algebra and Linear Models (2nd ed.)Berberian: Fundamentals of Real AnalysisBlyth: Lattices and Ordered Algebraic StructuresBoltyanskii/Efremovich: Intuitive Combinatorial Topology. (Shenitzer, trans.)Booss/Bleecker: Topology and AnalysisBorkar: Probability Theory: An Advanced CourseBöttcher/Silbermann: Introduction to Large Truncated Toeplitz MatricesBridges/Vîţă: Techniques of Constructive AnalysisCarleson/Gamelin: Complex DynamicsCecil: Lie Sphere Geometry: With Applications to SubmanifoldsChae: Lebesgue Integration (2nd ed.)Charlap: Bieberbach Groups and Flat ManifoldsChern: Complex Manifolds Without Potential TheoryCohn: A Classical Invitation to Algebraic Numbers and Class FieldsCurtis: Abstract Linear AlgebraCurtis: Matrix GroupsDebarre: Higher-Dimensional Algebraic GeometryDeitmar: A First Course in Harmonic Analysis (2nd ed.)DiBenedetto: Degenerate Parabolic EquationsDimca: Singularities and Topology of HypersurfacesEdwards: A Formal Background to Mathematics I a/bEdwards: A Formal Background to Mathematics II a/bEngel/Nagel: A Short Course on Operator SemigroupsFarenick: Algebras of Linear TransformationsFoulds: Graph Theory ApplicationsFriedman: Algebraic Surfaces and Holomorphic Vector BundlesFuhrmann: A Polynomial Approach to Linear AlgebraGardiner: A First Course in Group TheoryGårding/Tambour: Algebra for Computer ScienceGoldblatt: Orthogonality and Spacetime GeometryGustafson/Rao: Numerical Range: The Field of Values of Linear Operators andMatricesHahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt GroupsHeinonen: Lectures on Analysis on Metric SpacesHolmgren: A First Course in Discrete Dynamical SystemsHowe/Tan: Non-Abelian Harmonic Analysis: Applications of SL(2, R)Howes: Modern Analysis and TopologyHsieh/Sibuya: Basic Theory of Ordinary Differential EquationsHumi/Miller: Second Course in Ordinary Differential Equations

(continued after index)

Douglas S. Bridges and Luminiţa Simona Vîţă

Techniques ofConstructive Analysis

Douglas S. Bridges Luminiţa Simona VîţăDepartment of Mathematics/Statistics Department of Mathematics/StatisticsUniversity of Canterbury University of CanterburyChristchurch, New Zealand Christchurch, New [email protected] [email protected]

Editorial Board(North America):

S. Axler K.A. RibetMathematics Department Mathematics DepartmentSan Francisco State University University of California at BerkeleySan Francisco, CA 94132 Berkeley, CA 94720-3840USA [email protected] [email protected]

Mathematics Subject Classification (2000): 03F60, 26E40, 46S30, 47S30, 03F55, 03F65, 68Q99

Library of Congress Control Number: 2006926441

ISBN-10: 0-387-33646-X e-ISBN-10: 0-387-38147-3ISBN-13: 978-0387-33646-6 e-ISBN-13: 978-0387-38147-3

Printed on acid-free paper.

© 2006 Springer Science+Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even ifthey are not identified as such, is not to be taken as an expression of opinion as to whether ornot they are subject to proprietary rights.

9 8 7 6 5 4 3 2 1

springer.com

Ideal pierdut ı̂n noaptea unei lumi ce nu mai este,Lume ce gândea ı̂n basme şi vorbea ı̂n poezii,O! te văd, te-aud, te cuget, tânără şi dulce vesteDintr-un cer cu alte stele, cu-alte raiuri, cu alţi zei.

—Mihai Eminescu, “Venere şi Madonă”

Oh, ideal lost in night-mists of a vanished universe:People who would think in legends—all a world who spoke in verse;I can see and think and hear you—youthful scout which gently nodsFrom a sky with different starlights, other Edens, other gods.

—Mihai Eminescu, “Venus and Madonna”(translated by Andrei Bantaş)

This image is a courtesy of The Times of London. Printed in the February 3,2004 issue.

Preface

Rosencrantz: Shouldn’t we be doing something... constructive?

Guildenstern: What did you have in mind?

—Tom Stoppard, Rosencratz and Guildenstern Are Dead

We have written this book in order to provide an introduction to constructive anal-ysis, emphasising techniques and results that have been obtained in the last twentyyears. The intended readership comprises senior undergraduates, postgraduates,and professional researchers in mathematics and theoretical computer science. Wehope that our work will help spread the message that doing mathematics construc-tively is interesting (it can even be fun!) and challenging, and produces new, deepcomputational information.

An appreciation of the distinction between constructive and nonconstructive hasbecome more widespread in this era of computers. Nevertheless, there are few booksdevoted to the development of mathematics in a rigorously constructive/computablefashion, although there are some, primarily concentrating on logic and foundations,in which the odd chapter deals with constructive mathematics proper as distinctfrom its underlying logic or set theory. It is now almost forty years since the publi-cation of Errett Bishop’s seminal monograph Foundations of Constructive Analysis[9], which in our view is one of the most remarkable intellectual documents of thetwentieth century, and more than twenty since the appearance of its outgrowth[12]. In the intervening years there has been considerable activity in constructiveanalysis, algebra, and topology; in related foundational areas such as type theory[69]; and in the relation between constructive mathematics and computer science(for example, program extraction from proofs [42, 70, 51]). Believing that a newintroduction to the mathematical, as distinct from the foundational, side of thesubject is overdue, we embarked upon this monograph.

Our book is intended not to replace, but to supplement, Bishop’s original classic[9] and the later volume [12] based thereon. Both of those two monographs cover

viii Preface

aspects of analysis, such as Haar measure and commutative Banach algebras, thatwe do not mention. We cover some topics that are found in [9] and [12] (it would bealmost inconceivable to produce a book like ours, dealing with constructive mathe-matics for nonexperts, without proving, for example, basic results about locatednessand total boundedness); but we have tried to provide improved proofs wheneverpossible. However, much of the material we present was simply not around at thetime of writing of [9] or [12].

Instead of systematically developing analysis, beginning with the real line andcontinuing through metric, normed, and Hilbert spaces to its higher reaches, wehave chosen to write the chapters around certain themes or techniques (hence ourtitle). For example, Chapter 3 is devoted to the λ-technique, which, since its first usein the proof of Lemma 7 on page 177 of [9], has become a surprisingly powerful toolwith applications in many areas of constructive analysis. A major influence in theapplication of the λ-technique was Ishihara’s remarkable paper [60], which showedthat a subtle use of the technique could enable us to prove disjunctions whose proof,although trivial with classical logic, appears at first sight to be constructively outof the question. This paper opened up many new pathways in constructive analysis.

Chapter 1 introduces constructive mathematics and lays the foundations forthe later chapters. In Chapter 2 we first present a new construction of the realnumbers, motivated by ideas in [2]. After deriving standard properties such as thecompleteness of R, we introduce metric spaces, with the major theme of locatedness,and normed linear spaces. When we discuss metric, normed, and Hilbert spaces, weassume some familiarity with the standard classical definitions of those conceptsand with those elementary classical properties that pass over unchanged to theconstructive setting.

Chapter 3 we have already referred to. The main theme of Chapter 4 is finite-dimensionality, but the chapter concludes with an introduction to Hilbert spaces.

Chapter 5 deals with convexity in normed spaces. Starting with some elementaryconvex geometry in Rn, the chapter goes on to handle separation and Hahn–Banachtheorems, locally convex spaces, and duality. Following Bishop, we describe thoselinear functionals that are weak∗-uniformly continuous on the unit ball of the dualspace. We then give a new application of the technique used to prove that result,thereby characterising certain continuous linear functionals on the space of boundedoperators on a Hilbert space.

In Chapter 6 we derive a range of results associated with the theme of located-ness and with the λ-technique introduced in Chapter 3. We pay particular attentionto necessary and sufficient conditions for convex subsets of a normed space to belocated, and to connections between properties of an operator on a Hilbert spaceand those of its adjoint—when that adjoint exists: it may not always do so construc-tively. The final section of the book deals with a relatively recent version of Baire’stheorem and its applications, and culminates in constructive versions of three ofthe big guns in functional analysis: the open mapping, inverse mapping, and closedgraph theorems.

Preface ix

Which parts of the book deal with new material, compared with what appearedin [12]? We have already mentioned the new construction of the real numbers, inChapter 2. Notable novelties in the later chapters include all but one result inChapter 3 on the λ-technique; the section on convexity, Ishihara’s results on exactHahn–Banach extensions, and our characterisation theorem for certain continuouslinear functionals, all in Chapter 5; and virtually all of Chapter 6. Throughout thebook there are what we hope will be seen as improvements and simplifications ofproofs of many results that were given in [9] or [12].

What do we mean by “constructive analysis” in the title of this book? We donot mean analysis carried out with the usual “classical” logic within a framework,such as recursive function theory, designed to capture the concept of computability.In our view, such a notion of constructive has at least two drawbacks. First, byworking within, say, the recursive setting, it can make the mathematics look lesslike normal mathematics and much harder to read. Secondly, the recursive con-straint removes the possibility of other interpretations of the mathematics, such asBrouwer’s intuitionistic one [48]. Our approach, on the other hand, has neither ofthese features: the mathematics looks and reads just like the mathematics one isused to from undergraduate days, and all our proofs and results are valid in severalmodels. They are valid in the recursive model, in intuitionistic mathematics, and, webelieve, in any of the models for “computable mathematics” (including Weihrauch’sType Two Effectivity Theory [91], within which Andrej Bauer has recently found arealisability interpretation of constructive mathematics within Weihrauch’s theory[5]). They are also valid proofs in standard mathematics with classical logic. Forexample, our proof of the Hahn–Banach theorem (Theorem 5.3.3) is, as it stands,a valid algorithmic proof of the classical Hahn–Banach theorem. Moreover—andthis is one advantage of a constructive proof in general—our proof embodies analgorithm for the construction of the functional whose existence is stated in thetheorem. This algorithm can be extracted from the proof, and, as an undeservedbonus, the proof itself demonstrates that the algorithm is correct or, in computerscience parlance, “meets its specifications”.1

So how do we achieve all this? Simply by changing the logic with which we doour mathematics! Instead of using classical logic, we systematically use intuition-istic logic, which was abstracted by Heyting [52] from the practice of Brouwer’sintuitionistic mathematics. The remarkable fact is that every proof carried out withintuitionistic logic is fully constructive/algorithmic. (Is this the “secret on the pointof being blabbed” that appears in the epigraph to Bishop’s book?) Unfortunately,too few mathematicians outside the mathematical logic community are aware ofthis serendipity and dismiss both intuitionistic logic and constructive mathematicsas at best a marginal curiosity. This contrasts sharply with the theoretical com-puter science community, in which there is considerable knowledge of, and interestin, the computational power of intuitionistic logic.

1We do not carry out program-extraction from proofs in our book. For more on thistopic see [42, 51, 70].

x Preface

Reading constructive mathematics demands careful interpretation. A theoremin this book might look like a familiar one from classical analysis, but with morecomplicated hypotheses and proof. However, the statement of the theorem will bephrased so that the explicit algorithmic interpretation is left to the reader; and theadditional hypotheses will be necessary for a constructive proof, which will containalgorithmic information that is excluded from the classical proof by the latter’suse of principles outside intuitionistic logic. Consider, for example, the followingstatement:

(*) Let C be an open convex subset of a normed space X, let ξ ∈ C, andlet z ∈ X be bounded away from C. Then the boundary of C intersects thesegment [ξ, z] joining ξ and z.

This is trivial to prove classically; but to find/construct the (necessarily unique)point in which the boundary of C intersects [ξ, z] is a totally different matter. Theconstructive theorem (Proposition 5.1.5 below) requires us to postulate that theunion of C and its metric complement −C (the set of points bounded away fromC) be dense in X, and that X itself be a complete normed space. The constructiveproof, though elementary, requires some careful geometrical estimation that wouldbe supererogatory in the natural classical proof by contradiction. The benefit ofthat estimation and of the use of intuitionistic logic is that we could extract fromthe constructive proof an implementable algorithm for finding the point wherethe segment crosses the boundary. In turn, this would enable us to produce analgorithm for constructing separating hyperplanes and Hahn–Banach extensions oflinear functionals, under appropriate hypotheses.

We could have made the algorithmic interpretation of the constructive versionof (*) explicit by stating the proposition in this way:

There is a “boundary crossing algorithm” that, applied to the data consistingof (i) an open convex set C in a Banach space X such that C ∪−C is densein X, (ii) a point ξ of C, and (iii) a point z of −C, constructs the pointwhere the boundary of C intersects the segment [ξ, z] .

Even this is not really explicit enough. A full description of the data to which theboundary crossing algorithm applies would require explicit information about thealgorithms for such things as these: membership of C; the convergence of Cauchysequences in X; the computation, for given x in X and ε > 0, of a point y of C∪−Csuch that ‖x − y‖ < ε (and even the decision between the cases “y ∈ C” and “y ∈−C”); and so on. Such explicit description of algorithmic hypotheses would becomean ever greater burden on writer and reader alike as the book probed deeper anddeeper into abstract analysis. It is a matter of sound sense, even sanity, to unburdenourselves from the outset, relying on the reader’s native wit in the interpretationof the statements of our constructive lemmas, propositions, and theorems.

We should make it clear that we are not advocating the exclusive use of in-tuitionistic logic in mathematics. That logic is, we believe, the natural and right

Preface xi

one to use when dealing with the constructive content of mathematics. To abandonclassical logic in those fields (such as the higher reaches of set theory) where con-structivity is of little or no significance makes no sense whatsoever. Nevertheless,it is remarkable how much mathematics actually has what Bishop called “a deepunderpinning of constructive truth”.

Christchurch, New Zealand Douglas BridgesJanuary 2006 Luminiţa Simona Vı̂ţă

Acknowledgments

It is never easy to apportion thanks properly among the many who have contributedto this book either directly or by their support and encouragement at various stagesof our professional lives. We do, however, have special thanks for the followingpeople:

Cris Calude, who was responsible for bringing Vı̂ţă to New Zealand to begin whathas proved a very fruitful research partnership with Bridges, and who has beentireless in his encouragement of our work over many years.

Hajime Ishihara, who, in conjunction with one or both of the authors, was largelyresponsible for much of the work in Chapter 6 and whose influence can be seen inseveral other places in the book. (Hajime’s contributions to constructive functionalanalysis have been remarkable and deserve to be recognised more widely.) In addi-tion, he has hosted us many times at the Japan Advanced Institute of Science &Technology, each of our visits there being both memorable and highly productive.

Peter Schuster, not only for his contributions to constructive mathematics since hejoined our community ten years ago, but also for acting as organiser, fund-raiser,and host on our many research visits to Munich. His considerable efforts in securinga DAAD Gastprofessorship for Bridges in the Mathematisches Institut der Ludwig-Maximilians-Universität (LMU), München, in 2003 provided us with the time andenvironment in which we could break the back of the writing of this book.

Our early drafts of Chapters 1–5 formed the basis of graduate lectures by Bridgesat LMU in 2003. We thank the students in that course for patiently receiving thatmaterial and for suggesting corrections and improvements to our presentation of it.We are also grateful to

� the DAAD for supporting Bridges as a Gastprofessor at LMU for that year;� Otto Forster and Helmut Schwichtenberg, our hosts at LMU in 2003 and on

several other occasions;

xiv Acknowledgments

� the New Zealand Foundation of Research, Science and Technology for Vı̂ţă’spostdoctoral fellowship from 2002 to 2005 and for supporting her extendedvisit to LMU to enable us to work together on the book;

� our departments at Canterbury and Galaţi;� Josef Berger, Hannes Diener, Maarten Jordans, and Robin Havea, who have

kindly assisted us with proofreading various chapters.

We say a warm “thank you” to our friends Imola and Attila Zsigmond, and Helmutand Eva Pellinger, who were wonderful hosts during our time in Munich.

Finally, we want to thank our families for their continuing love and support.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Introduction to Constructive Mathematics . . . . . . . . . . . . . . . . . . . . . 1

1.1 What Is Constructive Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 A Very Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Intuitionistic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Informal Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Techniques of Elementary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 The Real Number Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 The λ-Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.1 Introduction to the Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Ishihara’s Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Finite-Dimensional and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1 Finite-Dimensional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Best Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Linearity and Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

xvi Contents

5.1 Crossing Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2 Separation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3 The Hahn–Banach Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.4 Locally Convex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6 Operators and Locatedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.1 Smooth and Uniformly Smooth Normed Spaces . . . . . . . . . . . . . . . . . 147

6.2 Locatedness of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.3 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.4 Functions of Selfadjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.5 Locating the Kernel and the Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.6 Baire’s Theorem, with Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

1

Introduction to Constructive Mathematics

My task which I am trying to achieve is, by the power of the written word, tomake you hear, to make you feel—it is, before all, to make you see. That, and nomore, and it is everything.

—Joseph Conrad, The Nigger of the ‘Narcissus’

In this chapter we first sketch the history and philosophy that motivated the early workersin the field of constructive mathematics. We then describe informal intuitionistic logicand discuss a number of elementary classical theorems that do not carry over to theconstructive setting. Finally, we introduce an informal constructive theory of sets andfunctions. All this will prepare us for the presentation of the constructive theory of thereal line R in Chapter 2, and for the more abstract analysis that will be described in laterchapters.

1.1 What Is Constructive Mathematics?

Proposition 20 in Book IX of the thirteen volumes of Euclid’s Elements states that

Prime numbers are more than any assigned multitude of prime numbers

—in current terms, there are infinitely many primes. The modernised version ofEuclid’s proof is often presented as follows. Suppose that there are only finitelymany primes, say p1, . . . , pn, and consider the integer

p = p1 × p2 × · · · × pn + 1.Being greater than 2, p has prime factors (it may even be prime itself). Since thenumbers pk are not divisors of p, each prime factor of p is distinct from each pk.This is absurd, since {p1, . . . , pn} is supposed to be the set of all primes. From thiscontradiction we conclude that the set of primes is infinite.

2 1 Introduction to Constructive Mathematics

Although at one level there appears to be nothing untoward about this proof,it can be criticised on two counts. First, it uses a totally unnecessary contradictionargument. If you look carefully, you will see that the proof actually embodies analgorithm that, applied to any finite set {p1, . . . , pn} of primes, enables you tocompute a prime that is distinct from each element of that set. In other words,the use of a contradiction argument in the preceding paragraph has obscured thecomputational content of Euclid’s proof.

The second criticism of the proof is a little more subtle, and deals with thenotion of “infinitely many”. The proof is based on the negative idea that a set isinfinite if and only if it is contradictory that it be finite. But an algorithmic re-casting of Euclid’s proof, as suggested in the preceding paragraph, shows that theset S of primes is infinite in a more positive, productive sense: namely, if we startwith a finite subset F of S, then we can compute an element of S that is distinctfrom each element of F.

From a traditional standpoint, the distinctions between the contradiction proofof Euclid’s theorem and the algorithmic one, and between the negative and positivenotions of “infinite”, are obscured if not invisible. For example, the two notions of“infinite” are equivalent if we use traditional logic—or classical logic, as it is nor-mally called—so the distinction has to be perceived at an aesthetic level rather thana mathematical one. The same applies, more generally, to a proof by contradictionof the existence of an object x with the property P (x). In such a proof one sup-poses that P (x) is false for all applicable objects x, deduces a contradiction, andthen concludes that P (x) must, after all, hold for some x, even though the proofdoesn’t tell us which x actually has the desired property. Classical logic draws nodistinction between the “idealistic existence” demonstrated by such a proof and the“constructive existence” based on an algorithm that constructs x and shows thatP (x) holds. In order to reveal such distinctions at a mathematical, rather than anaesthetic, level we shall adopt the radical expedient of changing our logic: through-out this book, we shall work with intuitionistic logic, an abstraction of the informallogic used in algorithmic thinking.

How much analysis, as normally presented, is really nonconstructive —that is,essentially dependent on proofs by contradiction or other nonalgorithmic proce-dures? Consider, for example, the classical intermediate value theorem:

If f : [a, b] −→ R is a continuous mapping such that f(a) < 0 and f(b) > 0,then there exists c ∈ (a, b) such that f(c) = 0.

(Note that [a, b] and (a, b) respectively denote the closed and open intervals withendpoints a and b. We shall use standard notations, like (a, b] for the half-openinterval, without further comment.)

It might be thought that the common elementary proofs of the intermediatevalue theorem are constructive, enabling one to produce a zero c of the function f.For example, one proof uses interval-halving in the following way. Without loss of

1.1 What Is Constructive Mathematics? 3

generality, take a = 0 and b = 1. Consider f(1/2): if it is 0, then we take c = 0and stop the process; if f(1/2) > 0, then f satisfies the hypotheses of the theoremwith a = 0 and b = 1/2; if f(1/2) < 0, then f satisfies the hypotheses with a = 1/2and b = 1. In each of the last two cases, we proceed with the interval-halving. Thisprocess either stops after a finite number of iterations and produces the requiredzero of f, or else it goes on ad infinitum to produce a descending sequence ofcompact intervals whose unique point of intersection is the required zero. Isn’t thisa fully algorithmic proof?

Suppose we try to implement the algorithm embodied in this proof on a com-puter that works with 50-bit precision. What happens if we apply it to the cubicfunction f defined on [0, 1] by

f(x) =(

x − 34

)(x − 1

2

)2− 2−51?

Here,

f(0) = − 316

− 2−51 < 0, f(1) = 116

− 2−51 > 0,so we are well set to carry out the first step of the interval-halving algorithm. Sinceour computer’s floating-point representation of f(1/2) is 0 (we have the phenom-enon of underflow, in which the computer sets the small but nonzero number −2−51equal to 0), the algorithm stops by outputting c = 1/2 as the place where f has azero. But in this case the only zero of f in [0, 1] lies between 3/4 and 1, more thanone quarter of the entire interval away from the output value 1/2.

Now, one could object that this example is misleading, in that the problemarises from the level of precision in the computer rather than any intrinsic failingin the algorithm itself. To deal with this point, for each positive integer n let G(n)signify that 2n + 2 is a sum of two primes. Construct a binary sequence (an)n�1such that for each n,

an = 0 if and only if either G(k) for all k � n or else there exists k < nsuch that ¬G(k).

Define a =∞∑

n=1an2−n, and note that a = 0 if and only if the Goldbach conjecture,

Every even integer greater than 2 is a sum of two primes,

holds. Using classical logic, apply the classical interval-halving algorithm to thecubic function f defined on [0, 1] by

f(x) =(

x − 34

)(x − 1

2

)2− a.

As long as the status of the Goldbach conjecture remains undecided (which it hasdone since the conjecture first appeared in 1742), no matter what finite precision


our computer has, the algorithm will output 1/2 as a zero of f ; but if the Goldbachconjecture is false, then f(1/2) = −a < 0 and the first zero of f in [0, 1] occursbetween 3/4 and 1. In fact, the classical algorithm will give the correct output ifand only if the Goldbach conjecture is true. However, as the reader may verify, aconstructive approximate interval-halving argument, such as that expected in thesolution of Exercise 11, does not give the possibly false value 1/2 for a zero of f.It produces a value that approximates the zero of f lying between 3/4 and 1, asaccurately as the precision of the computer permits.

In this example, the classically (but not constructively) defined function takingthe parameter a to the smallest root r(a) of f is discontinuous at a = 0. Theclassical algorithm correctly outputs r(a) = 1/2 in the case a = 0; but if a > 0,then, by outputting the value 1/2, the algorithm has failed to spot that the value ofr(a) jumps from 1/2 to more than 3/4 as the parameter a increases from 0. Thereis a general principle that constructive proofs will involve continuity in parameters.Thus we cannot expect to prove constructively that for each a the above cubicfunction f has a smallest zero.

The problem with the classical interval-halving algorithm is that the finite pre-cision of the computer prevents it from making correct comparisons between twovery close, but distinct, real numbers. If we are to develop mathematics in a compu-tational manner, we have to ensure that such comparisons are barred. This barringcan be done in at least two ways. One way is to use classical logic and to precludenonalgorithmic “decisions” (such as whether two given numbers are equal) by devel-oping the mathematics using a standard programming language or a more abstractalgorithmic framework like that of recursive function theory. Another way is tochange from classical to intuitionistic logic. The advantages of this second way are,first, that nonalgorithmic “decisions” are automatically barred by the logic, and,second, that the resulting mathematics looks like the mathematics we are used tofrom school and university, without any special logical notation such as is used in,for example, recursive function theory.

In this book we explore mathematics with intuitionistic logic. We work through-out with notions, like that of “infinitely many” discussed earlier, that have positivecomputational meaning; and we present only algorithmic proofs—ones that showhow we can, at least in principle, construct the objects whose existence is assertedin the statement of a theorem. We hope to convince the reader that, contrary to awidely held belief, intuitionistic logic suffices for the development of deep, interest-ing mathematics and often opens up new vistas that are hidden by classical logic.In other words, we want to justify our belief in the power of positive (constructive)thinking in mathematics.

1.2 A Very Brief History 5

1.2 A Very Brief History

Although luminaries such as Leopold Kronecker had advocated a constructive ap-proach to mathematics in the nineteenth century, the story of modern construc-tivism really begins with the publication, in 1907, of the doctoral thesis “Onthe Foundations of Mathematics” [41], in which the Dutch mathematician L.E.J.Brouwer introduced his intuitionistic mathematics (INT) as an alternative to tra-ditional classical mathematics (CLASS). According to Brouwer, mathematical ob-jects are free creations of the human mind, independent of both logic and language,and a mathematical object comes into existence precisely when it is constructed.Such a belief naturally leads to a rejection of existence proofs by contradiction, anda consequent scepticism about the meaning of many of the theorems of CLASS. Notsurprisingly, Brouwer’s views met with at best indifference, and at worst hostility,from the large majority of his peers, for whom the elimination of nonconstructivearguments, with all their apparent power and fruitfulness, was too great a price topay for a clarification of the meaning of mathematics.

If we adhere to the principle that “existence” should always be interpretedconstructively, then we are forced to dispense with the unrestricted use of thelogical law of excluded middle (or excluded third),

P or (not P ),

which we shall abbreviate to LEM. Recognising this consequence of his philosophicalviews, Brouwer went as far as to claim,

The belief in the universal validity of the principle of the excluded third in math-ematics is considered by the intuitionists as a phenomenon of the history of civ-ilization of the same kind as the former belief in the rationality of π, or in therotation of the firmament about the earth [44].

Subsequently, he introduced into INT some principles that led to results ap-parently contradicting aspects of classical mathematics. For example, Brouwer wasable to prove that any real-valued function on [0, 1] is uniformly continuous. Butto regard Brouwer’s mathematics as inconsistent with its classical counterpart is aserious oversimplification of the situation, since the two types of mathematics arein many respects incomparable. Nevertheless, there was, and remains, a commonlyheld belief that too much mathematics has to be given up in order to accommodateBrouwer’s ideas. For example, Hilbert expressed his disagreement with Brouwer inwords both forceful and memorable:

Forbidding a mathematician to make use of the principle of excluded middle islike forbidding an astronomer his telescope or a boxer the use of his fists [54].

Despite continuing opposition, intuitionism survived and new constructive ap-proaches to mathematics arose. In 1948–1949 in the former Soviet Union, A.A.


Markov initiated a programme of recursive constructive mathematics (RUSS)—mathematics using intuitionistic logic and based on the Church–Markov–Turingthesis that all computable partial functions from the set N of natural numbers toitself are recursive. This approach led to a number of technical successes [66, 67].RUSS does not use any of Brouwer’s nonlogical intuitionistic principles; indeed, itcould not, since it produces results that are false if interpreted directly within INT.For example, in RUSS there exists a continuous real-valued map on [0, 1] that is notuniformly continuous; more dramatically, there exists a uniformly continuous mapf from [0, 1] onto (0, 1] that has infimum equal to 0. Once again, one should notoverreact to the apparent conflict with classical mathematics: the last of these re-sults should really be interpreted as saying that there exists a recursively uniformlycontinuous recursive function f from the closed interval [0, 1] of the recursive realline onto the recursive interval (0, 1] that has infimum equal to 0. Put this way,the result does not conflict with CLASS; indeed, it is a result of CLASS, sincethe proof within RUSS is actually a proof within CLASS that does not use suchnonconstructive logical principles as LEM.

By the mid-1960s, constructive mathematics was, when compared with its clas-sical counterpart, virtually stagnant. The situation changed in 1967 with the pub-lication of Errett Bishop’s monograph Foundations of Constructive Mathematics[9]. This book and its offspring [12] represent the most far-reaching and systematicpresentation of constructive analysis to date. In [9], Bishop revealed, by thorough-going constructive means but without resorting to either Brouwer’s principles orthe formalism of recursive function theory, a vast panorama of constructive mathe-matics, covering elementary analysis, metric and normed spaces, abstract measureand integration, the spectral theory of selfadjoint operators on a Hilbert space,Haar measure, duality on locally compact groups, and Banach algebras. Bishop’sconstructive mathematics (BISH) was founded on a primitive, unspecified notion of“algorithm”, or “finite routine”, and the Peano properties of natural numbers, andkept strictly to the interpretation of “existence ” as “computability”. His refusal topin down the notion of algorithm led to criticism, particularly from philosophers ofmathematics and from those committed to the Church–Markov–Turing thesis; butthis very imprecision enabled Bishop’s work to have a variety of interpretations: hisresults are valid in CLASS, INT, RUSS, and all reasonable models of computablemathematics, such as the more recent one propounded by Weihrauch [91]. Indeed,from a purely formal viewpoint, each of INT, RUSS, and CLASS can be regardedas BISH plus some additional principles: INT can be regarded as BISH supple-mented by Brouwer’s continuity principle and fan theorem; RUSS as BISH plusthe Church–Markov–Turing thesis; and CLASS as BISH plus the law of excludedmiddle.

One consequence of this multiplicity of interpretations is that we can oftendemonstrate that certain propositions P are independent of BISH; that is, neitherP nor (not P ) can be proved within BISH. For example, since “every mappingfrom [0, 1] into R is uniformly continuous” is a theorem of INT, and “there existsa continuous map of [0, 1] into R that is not uniformly continuous” is a theorem of

1.3 Intuitionistic Logic 7

RUSS, and since both INT and RUSS are formally consistent with BISH, withinBISH we cannot expect either to prove that every continuous map of [0, 1] into Ris uniformly continuous or to construct an example of a real-valued function thatis defined, but not uniformly continuous, on [0, 1].

Over the years since the publication of Bishop’s book, it became clear to a num-ber of researchers that, in essence, BISH is simply mathematics with intuitionisticlogic together with some appropriate set-theoretic foundation. As we pointed outat the end of the preceding section, working with intuitionistic logic automaticallybars noncomputational steps. As long as we keep strictly to intuitionistic logic,having made sure that our set-theoretic principles do not inadvertently imply LEMor some other nonconstructive proposition, the mathematics we develop turns outto be predictive, in the sense that every proof implicitly shows that if we performcertain calculations, we shall achieve certain results. Accordingly, when we speakof “constructive mathematics” or “BISH” in future, we shall mean “mathematicswith intuitionistic logic”. It therefore behooves us to explain more clearly exactlywhat intuitionistic logic is.

1.3 Intuitionistic Logic

The meaning doesn’t matter if it’s only idle chatter of a transcendental kind.

—W.S. Gilbert, Patience

Everywhere one seeks to produce meaning, to make the world signify, to renderit visible.

—Jean Baudrillard, Seduction, or the Superficial Abyss

For Brouwer, mathematics took precedence over logic. In order to describe the logicused by the (intuitionist) mathematician, it was necessary first to analyse the math-ematical processes of the mind, from which analysis the logic could be extracted.In 1930, Brouwer’s most famous pupil, Arend Heyting (1898–1980), published aset of formal axioms that so clearly characterise the logic used by the intuitionistthat they have become universally known as the axioms for intuitionistic logic [52].These axioms capture the so-called BHK interpretation of the connectives

∨ (or), ∧ (and), =⇒ (implies), ¬ (not)

and quantifiers∃ (there exists), ∀ (for all/each),

which we now outline.


� P ∨ Q : either we have a proof of P or else we have a proof of Q.� P ∧ Q : we have both a proof of P and a proof of Q.� P =⇒ Q : by means of an algorithm we can convert any proof of P into a proof

of Q.

� ¬P : assuming P, we can derive a contradiction (such as 0 = 1); equivalently,we can prove (P =⇒ (0 = 1)) .

� ∃x P (x) : we have (i) an algorithm that computes a certain object x, and (ii) analgorithm that, using the information supplied by the application of algorithm(i), demonstrates that P (x) holds.

� ∀x ∈ AP (x) : we have an algorithm that, applied to an object x and a proofthat x ∈ A, demonstrates that P (x) holds.

Note that in the interpretation of the statement ∀x ∈ AP (x), the proof ofP (x) will normally use both the data describing the object x and the informationsupplied by a proof that x belongs to the set A. This is an important point, sinceupon it hinges a key argument against the use of the axiom of choice in constructivemathematics. We shall return to this matter later.

A property P (x) is said to be decidable if for each x to which it might beapplicable we have

P (x) ∨ ¬P (x),where the disjunction and negation are given their BHK interpretations. Even fora decidable property P (n) of natural numbers n the property

∀nP (n) ∨ ¬∀nP (n),

and hence a fortiori LEM, will not hold in general. As a result, many classicalresults cannot be proved constructively, since they would imply LEM or perhapssome other manifestly nonconstructive principle.

To illustrate this point, consider the following simple statement, the limitedprinciple of omniscience (LPO):

∀a ∈ {0, 1}N+ (a = 0 ∨ a = 0) ,

where a = (a1, a2, . . .) , N+ = {1, 2, . . .} is the set of positive integers, {0, 1}N+

isthe set of all binary sequences, and

a = 0 ⇐⇒ ∀n (an = 0) ,a = 0 ⇐⇒ ∃n (an = 1) .

In words, LPO states that for each binary sequence (an)n�1, either an = 0 for alln or else there exists n such that an = 1. Of course, this is a triviality from the

1.3 Intuitionistic Logic 9

viewpoint of classical logic. But its BHK interpretation is not so simple: it saysthat there is an algorithm that, applied to any binary sequence a, either verifiesthat all the terms of the sequence are 0 or else computes the index of a term equalto 1. Anyone familiar with computers ought to be highly sceptical about such analgorithm, since in the case a = 0 it would normally need to test each of theinfinitely many terms an in order to come up with the correct decision.

For such reasons we feel justified in not accepting LPO, or any classical propo-sition that constructively implies LPO, as a valid principle of constructive mathe-matics. But we have another reason for not doing so: it can be shown that thereare models of Heyting arithmetic—Peano arithmetic with intuitionistic logic—inwhich LPO is false; so LPO cannot be derived in Heyting arithmetic (see [34, 48]).Since LPO is a special case of the law of excluded middle, we are led, in turn, torenounce the latter when working constructively. Similar informal analyses lead usto exclude both the classical rule

¬¬P =⇒ P,which forms the basis of proof by contradiction, and the following lesser limitedprinciple of omniscience (LLPO), which is easily seen to be a consequence of LPO:

For each binary sequence a with at most one term equal to 1 (in the sensethat aman = 0 for all distinct m and n), either a2n = 0 for all n or elsea2n+1 = 0 for all n.

The exclusion of such principles from constructive mathematics has serious con-sequences for mathematical practice. For example, we cannot hope to prove con-structively the simple statement

∀x ∈ R (x = 0 ∨ x = 0) , (1.1)where R denotes the set of real numbers and x = 0 means that we can computea rational number strictly between 0 and x (which, as we shall see when we dealwith the real numbers more formally in Chapter 2, is not the same, constructively,as proving that ¬ (x = 0)). To prove this, consider any binary sequence a, and useit to define the binary expansion of a real number

x =∞∑

n=1

an2−n.

If x = 0, then a = 0. If x = 0, we can compute a positive integer N such that

x > 2−N =∞∑

n=N+1

2−n;

it is then clear that, by testing the terms a1, . . . , aN , we can find n � N such thatan = 1. Thus statement (1.1) about real numbers implies LPO and is thereforeessentially nonconstructive.


If the binary sequence a has at most one term equal to 1, then we can use thereal number ∞∑

n=1

(−1)n an2−n

to show that the statement

∀x ∈ R (x � 0 ∨ x � 0)

implies LLPO.

The following elementary classical statements also turn out to be nonconstruc-tive.

� Each real number x is either rational or irrational (that is, x = r for eachrational number r). To see this, consider

x =∞∑

n=1

1 − ann!

,

where a is any increasing binary sequence (that is a binary sequence such thatan � an+1 for each n).

� Each real number x has a binary expansion. Note that the standard interval-halving argument for “constructing” binary expansions does not work, sincewe cannot necessarily decide, for a given number x between 0 and 1, whetherx � 1/2 or x � 1/2. In fact, the existence of binary expansions is equivalent toLLPO.

� The intermediate value theorem, which is equivalent to LLPO.

� For all x, y ∈ R, if xy = 0, then either x = 0 or y = 0. The constructive failureof this proposition clearly has implications for the theory of integral domains.

We emphasise here that classically valid statements like “each real number is eitherrational or irrational” that imply omniscience principles are not false in constructivemathematics; they cannot be, since BISH is consistent with CLASS.

One principle whose constructive status is controversial is Markov’s principle(MP):

∀a ∈ {0,1}N+ (¬ (a = 0) =⇒ a = 0) ;in words, for any binary sequence a, if it is impossible for all the terms to equal0, then there exists a term equal to 1. In order to accept this as a principle ofconstructive mathematics, you have to be convinced that the information conveyedby the antecedent ¬ (a = 0) is sufficient to enable us to compute an index n withan = 1. The argument in favour of MP says that we can carry out this computationby searching systematically through the terms an, since the hypothesis ¬ (a = 0)

1.4 Informal Set Theory 11

guarantees that we shall eventually stumble across a term equal to 1. The counter-argument is that the antecedent provides us with no prior bound for such a search—it does not tell us how many terms we need to test before we arrive at one equal to1—so the search might go on longer than the remaining life of the universe beforeit produced the desired result. Moreover, Markov’s principle, like LPO, cannot beproved within Heyting arithmetic. For these reasons, we shall follow the normalpractice of excluding MP from the working principles of constructive mathematics.As a consequence we exclude the even stronger logical principle

(∀x ∈ A (P (x) ∨ ¬P (x)) ∧ ¬∀x ∈ A¬P (x)) =⇒ ∃x ∈ AP (x), (1.2)where A is a well-defined set (the exact meaning of “well-defined set” will becomeclear in the next section). In fact, even if we were to accept Markov’s principle onthe grounds that an unbounded search through the natural numbers that cannotfail to terminate must eventually do so, we would balk at accepting (1.2), since fora general set A there will be no natural order allowing us to search systematicallyin the way we can with N.

An example of the type dealt with earlier, in which a classically valid propositionP is shown constructively to entail an essentially nonconstructive principle likeLEM, LPO, LLPO, or even MP, is called a Brouwerian counterexample to P (eventhough it is not a counterexample in the true sense of the word; it is merely anindication that P does not admit of constructive proof). There is another expression

that we may use in this context. For example, we refer to the number x =∞∑

n=1an2−n

that we constructed from a given binary sequence (an)n�1 and then used to showthat (1.1) implies LPO as a Brouwerian example of a real number x for which wecannot decide whether x = 0 or x = 0.

1.4 Informal Set Theory

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.

—L. Kronecker [90]

The primary concern of mathematics is number, and this means the positiveintegers. We feel about number the way Kant felt about space. The positiveintegers and their arithmetic are presupposed by the very nature of our intelligenceand, we are tempted to believe, by the very nature of intelligence in general. Thedevelopment of the positive integers from the primitive concept of the unit, theconcept of adjoining a unit, and the process of mathematical induction carriescomplete conviction. In the words of Kronecker, the positive integers were createdby God.

—Errett Bishop [9]

Building on the set of positive integers and using intuitionistic logic, we followBishop’s approach to developing constructive mathematics at higher and higher


levels of abstraction. To do this, we need to clarify notions such as “set” and “func-tion”.

For us, a set X (other than our basic, primary set N+ of positive integers) isgiven by two pieces of data:

� a property that enables members of X to be constructed using objects that havealready been constructed (note this last phrase, which rules out the possibilityof impredicative definitions and therefore of Russell-type paradoxes), and

� an equivalence relation =X of equality between members of X.

We write x ∈ A to signify that x is an element of the set A, and x /∈ A instead of¬ (x ∈ A) .

The use of equivalence relations rather than intensional equality—that is, iden-tity of description—is common, but often goes unnoticed, in classical mathematics.For example, we call the rational numbers 1/2 and 3/6 equal, even though, strictlyspeaking, they are equivalent and not intensionally identical.

A subset S of a set X consists of a collection of elements drawn from X, togetherwith the equality relation induced on S by the given equality on X; that is, forelements x, y of S, we define

x =S y ⇐⇒ x =X y.We write S ⊂ T to signify that S is a subset of T. If P (x) is a property applicableto certain elements x of a set A, then we denote by

{x : x ∈ A ∧ P (x)}or

{x ∈ A : P (x)}the subset of A consisting of those elements x of A with the property P (x).

The logical complement of a subset S of X is

¬S = {x ∈ X : x /∈ S} .A particular example of this is the empty subset of X, defined by

∅X = ¬X.We say that a subset S of X is inhabited if

∃x (x ∈ S) .We then write S = ∅X . Note that in order to show that S is inhabited, we cannotjust prove that it is impossible for S to be empty; we must actually construct anelement of S; see Exercise 2.


Two sets X,Y are said to be equal sets if each is a subset of the other; in otherwords, if the sets have the same elements and the same equality relation.

We need to be careful when constructing new sets from old. Since an equality ispart of the data for a set, it does not make sense to talk of the union S ∪ T of twosets unless we can put together not only the sets as collections of objects but also,in some way, their given equality relations. In practice, this means that in order toconstruct their union S ∪ T, the sets S and T must be given as subsets of some setX. We then define

S ∪ T = {x ∈ X : x ∈ S ∨ x ∈ T},where the equality on S ∪ T is that induced by X. Likewise, the intersection of Sand T is defined only when S and T are subsets of some set X, and is then thesubset

S ∩ T = {x ∈ X : x ∈ S ∧ x ∈ T}of X.

The (Cartesian) product of two sets X,Y is the set X × Y consisting of allordered pairs (x, y) with x ∈ X and y ∈ Y, together with the equality given by

((x, y) =X×Y (x′, y′)) ⇐⇒ (x =X x′ ∧ y =Y y′) .

In many situations—even, as we shall see, on the real line—we frequently needa set X to be equipped with an inequality relation =X describing what it meansfor two elements of X to be unequal, or distinct. Such a relation must satisfy thefollowing two properties:

x =X y =⇒ ¬ (x =X y) ,x =X y =⇒ y =X x.

If, in addition,¬ (x =X y) =⇒ x =X y,

we say that the inequality is tight. A set X with an inequality is discrete if, for anytwo elements x and y of X, either x =X y or x =X y; we then also describe theinequality itself as discrete.

One inequality relation, the denial inequality, is defined by setting x =X y ifand only if ¬ (x =X y) . This inequality is normally too weak for practical purposes.For example, in the absence of Markov’s principle, on the real line R the property¬ (x =R 0) is weaker than |x| > 0 (Exercise 3); for that reason we define the standardinequality on R to be not the denial inequality but the one given by

x =R y ⇐⇒ |x − y| > 0.

From now on, when the meaning is clear from the context, we write

=, ∅, =, . . .


rather than=X , ∅X , =X , . . . .

A subset S of a set X with an inequality has a complement, defined by

∼S = {x ∈ X : ∀s ∈ S (x = s)} .Then ∼S ⊂ ¬S; but unless the inequality on X is the denial inequality, the reverseinclusion will not hold.

A subset S of a set X is said to be detachable (from, or in, X) if

∀x ∈ X (x ∈ S ∨ x /∈ S) .Since statement (1.1) implies LPO, not even the singleton subset {0} is detachablefrom R. However, {0} is detachable in the set Q of rational numbers.

When X comes with an inequality relation, we define the inequality on anysubset S of X to be the one induced by that on X:

x =S y ⇐⇒ x =X y.If also Y has an inequality relation, we define the inequality on the Cartesianproduct X × Y by

((x, y) =X×Y (x′, y′)) ⇐⇒ (x =X x′ ∨ y =Y y′) .

It should be no surprise that we require functions to be given by algorithms andto respect equality. Thus a function f from a set X to a set Y —also called a mapor mapping of X into Y , and written f : X −→ Y —is an algorithm that, appliedto any element x of X, produces an element f(x) of Y such that f is extensional :

∀x ∈ X ∀x′ ∈ X (x =X x′ =⇒ f(x) =Y f(x′)) .The element f(x) is called the value of f at x or the image of x under f . If Xand Y have inequality relations, then we may require f to be strongly extensional :

∀x ∈ X ∀x′ ∈ X (f(x) =Y f(x′) =⇒ x =X x′) .Note that the statement “all functions from R to R are strongly extensional” isequivalent to Markov’s principle.

Let f, g be two real-valued functions on a set X with an inequality. We say thatan element x0 of X is the strongly unique element of X such that f(x) = g(x) iff(x0) = g(x0) and f(x) = g(x) whenever x ∈ X and x = x0.Strong uniqueness will resurface in Chapter 4 in connection with best approxima-tions.

A partial function f : X −→ Y is a function from a subset of X into Y. Thesubset


{x ∈ X : f(x) is defined}is called the domain of f, denoted by dom (f) ; and the set

{y ∈ Y : ∃x ∈ X (y =Y f(x))}

the range of f, denoted by ran (f) . The image of a subset A of X under f is theset

f(A) = {f(x) : x ∈ A}.The inverse image of a subset B of Y under f is the set

f−1(B) = {x ∈ dom (f) : f(x) ∈ B} .

A partial function f : X −→ Y is said to be a total partial function on X ifdom(f) = X.

When the expression describing f(x) is given explicitly and the domain of thepartial function f is clearly understood, we may denote the function by x � f(x).For example, the partial function from R to itself whose value is defined, for eachreal number x such that x =R 0, to be 1/x may be written x � 1/x.

An important type of total partial function is defined as follows. Let X and Ibe sets. A family of elements of X with index set I (or indexed by I) is a mappingi � xi of I into X; we commonly denote this family by (xi)i∈I . In particular, if Iis N+, the family is called a sequence in X and is usually written (xn)n�1 . Moregeneral sequences of the form (xn)n�N have the obvious analogous meaning whenN is an integer.

Let f and g be mappings from subsets of a set X into a set Y, where Y isequipped with a binary operation �. We introduce the corresponding pointwiseoperation � on f and g by setting

(f�g)(x) = f(x)�g(x)

whenever f(x) and g(x) are both defined. Thus, taking Y = R, we see that the(pointwise) sum of f and g is given by

(f + g)(x) = f(x) + g(x)

if f(x) and g(x) are both defined; and that the (pointwise) quotient of f and g isgiven by

(f/ g)(x) = f(x)/ g(x)

if f(x) and g(x) are defined and g(x) = 0. If X = N+, so that f = (xn)n�1 andg = (yn)n�1 are sequences, then we also speak of termwise operations; for example,the termwise product of f and g is the sequence (xnyn)n�1.

Pointwise operations extend in the obvious ways to finitely many functions. Inthe case of a sequence (fn)n�1 of functions with values in a normed space (see


Chapter 2), once we have introduced the notion of a series in a normed space we

shall interpret∞∑

n=1fn in the obvious pointwise way.

A partial function f : X −→ Y can be identified with its graph,

G (f) = {(x, y) ∈ X × Y : x ∈ dom (f) , y ∈ ran (f) , y = f(x)},

a subset of the Cartesian product X×Y. We define two partial functions f : X −→ Yand g : X −→ Y to be equal if their graphs are equal as sets. Thus f, g are equalpartial functions if and only if their domains are equal subsets of X and f(x) = g(x)for each x ∈ dom(f).

We say that a partial function f : X −→ Y is

� one-one if f(x) = f(x′) entails x = x′;

� injective if X and Y are equipped with inequality relations, and x = x′ entailsf(x) = f(x′).

If f is injective and the inequality on X is tight, then f is one-one: for in that case,if f(x) = f(x′), then ¬ (x = x′) and so, by tightness, x = x′.

The composition, or composite, of partial functions f : A −→ B and g : B −→ Cis the partial function g ◦ f : A −→ C (sometimes written gf) defined by g ◦ f(x) =g(f(x)) wherever the right side exists.

A partial mapping f : X −→ Y maps its domain onto Y if

∀y ∈ Y ∃x ∈ X (y = f(x)) .

On the other hand, we say that f : X −→ Y is an epimorphism if there exists amapping g : Y −→ X such that

∀y ∈ Y (f (g(y)) = y) .

A one-one partial function f : X −→ Y has a one-one inverse

f−1 : ran (f) −→ dom (f)

defined by f−1 (f(x)) = x. If f is injective, then its inverse is strongly extensional.A bijection between X and Y is a one-one mapping from X onto Y .

The subtle distinction between mappings onto and epimorphisms is closelylinked to the constructive status of the axiom of choice, which we shall discussshortly. First, though, we introduce some notions of cardinality.

Let S be a subset of a set X with an inequality relation. We say that S is

� finitely enumerable if there exist a natural number N and a mapping of the set


{1, 2, . . . , N} = {n ∈ N+ : n � N}

onto S;

� finite if there exist a natural number N and an injective map of {1, 2, . . . , N}onto S.

When we speak of finitely many objects, we mean that those objects constitute aninhabited, finitely enumerable, but not necessarily finite, set.

Note that the case N = 0 of the definition of “finitely enumerable” showsthat the empty subset of X is finitely enumerable. Clearly, finite implies finitelyenumerable; the converse does not hold constructively.

We say that S is countable if there exist a detachable subset D of N+ and amapping φ of D onto S. Every finitely enumerable subset—in particular, the emptysubset—of X is countable. If D = N+, we call φ an enumeration of S, in whichcase we may denote S by {φ(1), φ(2), . . .} . If also φ is one-one, we say that Sis countably infinite. A set is countably infinite if and only if it is the range of aone-one mapping whose domain is a countably infinite, detachable subset of N+.

We now consider the axiom of choice (AC):

If X,Y are inhabited sets, S is a subset of X × Y, and for each x ∈ Xthere exists y ∈ Y such that (x, y) ∈ S, then there exists a choice functionf : X −→ Y such that (x, f(x)) ∈ S for each x ∈ X.

Under the BHK interpretation, the hypothesis

∀x ∈ X ∃y ∈ Y ((x, y) ∈ S)

of AC means that we have an algorithm that, applied to each element x of Xand the data showing that x belongs to X, constructs an element y of Y anddemonstrates that (x, y) ∈ S. This much is clear. However, there is no guaranteethat the algorithm will respect the equality relation on X—in other words, thatif x =X x′, and the algorithm constructs y, y′ in Y such that (x, y) ∈ S and(x′, y′) ∈ S, then y =Y y′. Indeed, we should expect that the computation of ymight use data that are associated with properties intrinsic to x that do not applyintrinsically to x′.

For example, anticipating our development of the real number set R, considerthe case in which X is R and Y is N+. A real number is (defined as) a certain set ofrational approximations. However, two equal real numbers x, x′ can have differentdefining sets of rational approximations. In that case, the algorithm that computes apositive integer n such that (x, n) ∈ S may, and in general will, compute a differentpositive integer n′ such that (x′, n′) ∈ S. These considerations throw real doubtover the possibility that there is a choice function implementing the algorithm.


In fact, an argument of Diaconescu [46] and Goodman & Myhill [50], but prefig-ured by Bishop (see Problem 2 on page 58 of [9]), shows that AC cannot be allowedas a principle of constructive mathematics.

Theorem 1.4.1. The axiom of choice implies the law of excluded middle.

Proof. Let P be any constructively meaningful statement, and define the set X toconsist of the two elements 0 and 1, together with the equality relation such that

(0 =X 1) ⇐⇒ P.Let Y be the set {0, 1} with the standard equality, and let S be the subset{(0, 0), (1, 1)} of the Cartesian product X × Y, taken with the standard equality.Suppose there exists a function f : X −→ Y such that (x, f(x)) ∈ S for all x ∈ X.There are three cases to consider: (i) f(0) = 1, (ii) f(1) = 0, and (iii) both f(0) = 0and f(1) = 1. In case (i) we have (0, 1) = (0, f(0)) ∈ S, so either (0, 1) =X×Y (0, 0)or (0, 1) =X×Y (1, 1). If the first of these two alternatives holds, then, by defini-tion of the equality on X × Y, we have 1 =Y 0, which is absurd. Hence, in fact,(0, 1) =X×Y (1, 1) . Thus, again by definition of the equality on X × Y, we have0 =X 1 and therefore P holds. Case (ii) similarly leads to the conclusion that Pholds. Finally, in case (iii) we have ¬ (f(0) =Y f(1)) ; therefore, since f is a func-tion, ¬ (0 =X 1) and so ¬P holds. Thus we have derived P ∨ ¬P from AC. �

The axiom of choice will hold constructively if the set X is one for which nocomputation is necessary to demonstrate that an element belongs to it; Bishop callssuch sets basic sets. Following the practice of most constructive mathematicians,we consider N+, the set

N = {0, 1, 2, 3, . . .}of natural numbers, and the set

Z = {0,±1,±2, . . .}of all integers to be basic sets. This practice is reflected in our acceptance of theprinciple of countable choice:

If Y is an inhabited set, S is a subset of N+ × Y, and for each positiveinteger n there exists y ∈ Y such that (n, y) ∈ S, then there is a functionf : N+ −→ Y such that (n, f(n)) ∈ S for each n ∈ N+.

In fact, many constructive proofs use the stronger principle of dependent choice:

If X is a set, a ∈ X, S is a subset of X × X, and for each x ∈ X thereexists y ∈ X such that (x, y) ∈ S, then there exists a sequence (xn)n�1 inX such that x1 = a and (xn, xn+1) ∈ S for each n ∈ N+.


Another contentious matter in constructive mathematics is the status and roleof the power set P (X) of a given set X: that is, the collection of all subsets of X,with equality of subsets as defined earlier. The main objection to admitting P (X)into the constructive fold is that we thereby allow impredicativity, since there isthen nothing to stop us constructing subsets of X whose defining characteristics areself-referential. On the other hand, nobody has yet shown that adding the powerset axiom

∀X∃Y ∀S (S ⊂ X ⇐⇒ S ∈ Y )to constructive mathematics enables us to prove LEM or some other incontestablynonconstructive principle.

There are ways of avoiding the power set. It often suffices to work with theset Y X of all mappings from X into a set Y. Since we have a clear idea of whatmappings from X into Y are (something we do not have for subsets of X), the setY X seems relatively innocent. Note that classically the set {0, 1}X can be identifiedwith the power set of X, since it comprises the characteristic functions of subsets ofX. This identification is not possible constructively, since characteristic functionsexist only for those subsets of X that are detachable.

Another way to avoid the full generality of the power set is to work with a well-defined but smaller set of subsets of X. For example, the set of compact subsets ofa metric space X is well defined (it is actually a metric space itself), and is oftenall we need for many parts of analysis.

Let S (X) be a well-defined set of subsets of X, with two elements taken as equalif and only if they are equal sets in the usual sense, and let I be some set. Then wecan speak sensibly about a family (Si)i∈I of elements of S (X) . We can also definethe union and intersection of such a family to be, respectively, the subsets⋃

i∈ISi = {x ∈ X : ∃i ∈ I (x ∈ Si)}

and ⋂i∈I

Si = {x ∈ X : ∀i ∈ I (x ∈ Si)}

of X. If I is the set of positive integers, we denote the above union and intersectionby

⋃n�1

Sn and⋂

n�1Sn, respectively.

The Cartesian product∏i∈I

Si is the subset of XI consisting of those functions f

such that f(i) ∈ Si for each i ∈ I; if X comes with an inequality relation, then thecorresponding inequality on

∏i∈I

Si is defined by

f = g ⇐⇒ ∃i ∈ I (f(i) = g(i)) .If I is the set of positive integers, then (Si)i∈I is a sequence of sets, and the elementsof∏i∈I

Si are also sequences; in this case, the sets Si can be arbitrary and need not


be subsets of a previously defined set X. If I = {1, 2, . . . , n}, then we denote theproduct

∏i∈I

Si by S1 × S2 × · · · × Sn, and refer to its elements (x1, x2, . . . , xn) asordered n-tuples, or, in the case n = 2, ordered pairs (something we used informallyin our earlier definition of the Cartesian product of two sets).

Other set-theoretic notions will be introduced later as they arise. It is now timeto turn our attention away from foundational matters to analysis proper.

Exercises

Although we shall not formally define the real number line R until Chapter 2, inthese exercises we assume elementary properties of real numbers where necessary.

1. Justify informally Brouwer’s observation that ¬¬¬P implies ¬P. Using this,show that the proposition (¬¬P =⇒ P ) is equivalent to the law of excludedmiddle.

2. Prove that if the statement

¬ (S = ∅) =⇒ S = ∅

applies to all subsets S of {0} , then the law of excluded middle holds.

3. Prove that∀x ∈ R (¬ (x = 0) =⇒ x = 0)

is equivalent to Markov’s principle.

4. Fill in the details of the proof that the statement

∀x ∈ R (x � 0 ∨ x � 0)

implies LLPO.

5. Give a Brouwerian counterexample to the proposition that every real numberx is either rational or irrational (where “x is irrational” means that x = r foreach rational number r).

6. Prove that every real number has a binary expansion if and only if LLPO holds.

7. Give a Brouwerian counterexample to the statement that if r1, r2, and r3 arereal roots of a quadratic polynomial x2 + ax + b with a, b ∈ R, then there existdistinct i, j with ri = rj .


8. Prove that the statement “for all real numbers x, y that have binary expansions,the sum x + y has a binary expansion” is equivalent to LLPO.

9. Prove that the statement

∀x, y ∈ R (xy = 0 =⇒ (x = 0 ∨ y = 0))is equivalent to LLPO.

10. Prove that the intermediate value theorem is equivalent to LLPO.

11. Let f : [a, b] −→ R be sequentially continuous in the following sense: for eachsequence (xn)n�1 in [a, b] that converges to a limit x, the sequence (f(xn))n�1converges to f(x). Suppose also that f(a)f(b) < 0, and that f is locally nonzeroin the sense that for each x ∈ [a, b] and each r > 0 there exists y ∈ [a, b]with |x − y| < r and f(y) = 0. Prove that there exists c ∈ (a, b) such thatf(c) = 0. (This version of the intermediate value theorem suffices for virtuallyall constructive purposes.)

12. Prove that the statement “all functions from R to R are strongly extensional”is equivalent to Markov’s principle.

13. Discuss the statement “every mapping from a set X onto a set Y is an epimor-phism”.

14. Give a Brouwerian counterexample to the statement “every subset of a finiteset is finitely enumerable”.

15. Prove that an inhabited set is countably infinite if and only if it is the range ofa function with domain N.

16. Prove that a subset S of N is countable if and only if there exists a sequence(Sn)n�1 of finite subsets of N such that S1 ⊂ S2 ⊂ S3 ⊂ · · · .

17. Prove that the statement “every inhabited subset of N is countable” impliesthe principle of finite possibility (PFP): to each binary sequence (an)n�1 therecorresponds a binary sequence (bn)n�1 such that an = 0 for each n if and onlyif there exists N such that bN = 1.

18. Prove that the intersection of two countable sets is countable.

19. Prove that the principle of dependent choice implies the principle of countablechoice. Prove also that the principle of dependent choice can be derived fromthe following principle of internal choice: if X is an inhabited set, S is a subsetof X × X, and for each x ∈ X there exists y ∈ X such that (x, y) ∈ S, thenthere exists a choice function f : X −→ X such that (x, f(x)) ∈ S for eachx ∈ X.


20. By a filter on a set X we mean a set F of inhabited subsets of X with thefollowing properties:

• X ∈ F.• If S and T belong to F, then S ∩ T ∈ F.• If S ∈ F and S ⊂ T ⊂ X, then T ∈ F.A filter U is called an ultrafilter if for all S ⊂ X, either S ∈ U or ¬S ∈ U. Theclassical ultrafilter principle states that every filter is contained in an ultrafil-ter. Prove that this principle implies the weak limited principle of omniscience(WLPO):

∀a ∈ 2N (∀n (an = 0) ∨ ¬∀n (an = 0)) .

Notes

The use of the Goldbach conjecture in the example in Section 1.1 is in no sense aconstraint on the argument; if the Goldbach conjecture were solved tomorrow, wecould easily replace it by any one of many open problems of number theory. Indeed,until Wiles proved the Fermat conjecture in 1994, it was common to use that, ratherthan the Goldbach conjecture, in such Brouwerian examples of nonconstructivity.

The reader interested in the history of foundations of mathematics should con-sult [44] and [84] for more information on the “Grundlagenstreit” that eventuatedbetween Brouwer and Hilbert in the 1920s. For more on Hilbert’s programme forthe secure foundation of mathematics see [93].

The designation “BHK interpretation” comes from Brouwer, Heyting, andKolmogorov. The BHK interpretation of implication, while more natural than theclassical one of material implication, in which (P =⇒ Q) is equivalent to (¬P ∨ Q),has not completely satisfied all researchers using constructive logic. Shortly be-fore he died, Bishop communicated to Bridges his dissatisfaction with the standardconstructive interpretation of implication. Unfortunately, he left nothing more thanvery rudimentary sketches of his ideas for its improvement. (Note, however, Bishop’spaper [10].) For a deeper analysis of the constructive interpretations of the connec-tives and quantifiers, we refer the reader to [48].

The classical invalidity of the recursive interpretation of LPO is not a matterof logic: it can be demonstrated, even with classical logic, that a recursive versionof LPO would lead to a proof of the decidability of the halting problem, which isknown to be impossible; see [34], pages 52–53.

Andrej Bauer has recently shown that proofs and results in BISH can be trans-lated into Weihrauch’s Type-2 Effectivity framework by a realisability interpreta-tion [5].


The principle (¬¬P =⇒ P ) is equivalent to LEM even with intuitionistic logic;see Exercise 1.

For the confirmed intuitionist there is at least one other reason for rejecting MP:it contradicts Brouwer’s (admittedly controversial) theory of the creating subject,an add-on to his intuitionistic mathematics. See [34] (pages 116–117) and, for arather different view, [73].

Bishop used “subfinite” instead of “finitely enumerable”. Note that a subset ofa finite set need not be finitely enumerable.

In the proof of Theorem 1.4.1 we could have defined X as a set of equivalenceclasses under the equivalence relation ∼ defined by

(0 ∼ 1) ⇐⇒ P,

but it is more in keeping with Bishop’s approach to proceed by considering a special,if unusual, equality relation on X.

Some constructive mathematicians argue against even the principle of countablechoice; see, for example, [76, 82].

Myhill has shown that, under the Church–Markov–Turing thesis, the power setof a singleton is uncountable: in other words, there is no recursive mapping of Nonto that power set [74] (page 364, Theorem 3). Within BISH we may not be able toprove the uncountability of the power set of {0} , but we certainly have an unendingsupply of subsets S of {0} for which we cannot decide that S = ∅ or S = {0}: givenany constructively meaningful statement P, we can define a corresponding set

S = {x : x = 0 ∧ P}

such that (S = {0} ∨ S = ∅) if and only if P ∨ ¬P.Myhill’s formal theory—based on primitive notions of “set”, “function”, and

“natural number”—is but one of several foundational theories advocated for con-structive mathematics. Others include Aczel’s constructive set theory [3], Martin-Löf’s type theory [69], and a largely unpublished constructive version of Morse’sset theory [16] in which membership of the universe appears to correspond to beingconstructively defined. For more on constructive foundational theories see [6] and[88].

2

Techniques of Elementary Analysis

My deplorable mania for analysis exhausts me.

—Gustave Flaubert, Letter (August 1846)

We begin by using a form of interval arithmetic as a foundation for the construction ofthe real number line R. Having discussed the elementary algebraic and order-theoreticproperties of real numbers, we prove that R is complete in two senses: Dedekind (order)complete and Cauchy (sequentially) complete. The next step is to generalise from thereals to metric and normed spaces. A particularly important property for us is totalboundedness, which plays a big part in proving the existence of suprema and infima; forthat reason we need to prove that there are lots of totally bounded subsets in a totallybounded space. We also highlight the important property of locatedness for subsets of ametric space, a property that holds automatically in classical mathematics.

2.1 The Real Number Line

Constructive analysis proper, as distinct from arithmetic, begins with the real num-bers, which we shall construct, choice-free, by interval arithmetic. First, though, weobserve that the purely algebraic constructions of the sets Z of integers and Q ofrational numbers from N are carried out as in classical algebra, and that the stan-dard inequality on Q —and hence a fortiori on Z regarded as a subset of Q —is the(in this case discrete) denial inequality.

By a real number we mean a subset x of Q × Q such that

� for all (q, q′) in x, q � q′;

� for all (q, q′) and (r, r′) in x, the closed intervals [q, q′] and [r, r′] in Q intersectin points of Q;

26 2 Techniques of Elementary Analysis

� for each positive rational ε there exists (q, q′) in x such that q′ − q < ε.

The last of these properties ensures that x is inhabited. It also legitimises our useof expressions like “pick an element of x”: to carry out such a selection, we simplytake ε = 1 in the third of the defining properties of the real number x, to producea corresponding element (q, q′) of x.

The intuition underlying our definition of “real number” is that the elementsof x are the rational endpoints of closed intervals with one point—namely x—incommon. Any rational number q gives rise to a canonical real number

q = {(q, q)}with which the original rational q is identified.

Two real numbers x and y are

• equal, written x = y, if for all (q, q′) ∈ x and all (r, r′) ∈ y, the intervals [q, q′]and [r, r′] in Q have a rational point in common;

• unequal (or distinct), written x = y, if there exist (q, q′) ∈ x and (r, r′) ∈ ysuch that the intervals [q, q′] and [r, r′] in Q are disjoint.

It is almost immediate that = satisfies the defining properties of an inequalityrelation. Let us check that equality is an equivalence relation. It is trivial that itis reflexive and symmetric, so only transitivity has to be handled. Let x = y andy = z, and suppose that for some (q, q′) ∈ x and (r, r′) ∈ z there are no rationalpoints in [q, q′]∩ [r, r′] . We may assume without loss of generality that q′ < r. Usingthe third of the defining properties for a real number, choose (s, s′) ∈ y such thats′ − s < r − q′. Now, the rational interval [s, s′] intersects [q, q′] in a rational pointu, and [r, r′] in a rational point v, so

r − q′ � v − u � s′ − s < r − q′,a contradiction. Hence

¬ ([q, q′] ∩ [r, r′] = ∅) .Since we are working with intervals in Q, with the aid of two simple lemmas wecan turn this around to construct a point of [q, q′] ∩ [r, r′] , and therefore completethe proof that x = z, as follows.

Lemma 2.1.1. Let a, b, c, d be rational numbers with a � b and c � d. Then thereexists a rational number in [a, b] ∩ [c, d] if and only if a � d and c � b.

Proof. If r is a rational number in the intersection of the intervals, then a � r � dand c � r � b, so a � d and c � b. If, conversely, these conditions hold, then eitherc < a and therefore a ∈ [a, b] ∩ [c, d] , or else a � c and c ∈ [a, b] ∩ [c, d] . �

2.1 The Real Number Line 27

Lemma 2.1.2. Let I, J be closed, bounded intervals in Q such that ¬ (I ∩ J = ∅) .Then there exists r ∈ Q such that r ∈ I ∩ J.

Proof. Let I = [a, b] and J = [c, d] . If b < c, then I ∩J = ∅, a contradiction. Hencec � b. Likewise, a � d. It remains to apply Lemma 2.1.1. �

Taken with the equality and inequality we have defined above, the collection ofreal numbers forms a set: the real line R.

Let x,y be real numbers. We say that x is greater than y, and that y is less thanx, if there exist (q, q′) ∈ x and (r, r′) ∈ y such that r′ < q; we then write x > y or,equivalently, y < x. On the other hand, we say that x is greater than or equal toy, and that y is less than or equal to x, if for all (q, q′) ∈ x and all (r, r′) ∈ y wehave q′ � r; we then write x � y or, equivalently, y � x. We write, for example,x > y to indicate that ¬ (x > y) , and x � y to indicate that ¬ (x � y) .

Clearly, x � x and x > x. Moreover, by Lemma 2.1.1, x = y if and only if bothx � y and y � x; and x = y if and only if either x > y or else x < y.

Lemma 2.1.3. If x > y, then y > x.

Proof. Since x > y, there exist (q, q′) ∈ x and (r, r′) ∈ y such that r′ < q. Ifalso y > x, then there exist (s, s′) ∈ x and (t, t′) ∈ y such that s′ < t. By thedefining conditions on real numbers, there exist rational numbers a, b such thata ∈ [q, q′] ∩ [s, s′] and b ∈ [r, r′] ∩ [t, t′] . But then

a � s′ < t � b � r′ < q � a,

a contradiction. �

Lemma 2.1.4. x � y if and only if y > x.

Proof. By definition, x � y if and only if for all (q, q′) ∈ x and all (r, r′) ∈ y wehave q′ � r, which occurs precisely when ¬(y > x). �

Lemma 2.1.5. If x > y, then x � y.

Proof. By Lemma 2.1.3, y > x. The result follows from Lemma 2.1.4. �

Lemma 2.1.6. If x > 0, then there exists a positive integer n such that x > 1/n.


Proof. First pick (q, q′) ∈ x such that 0 < q. Then choose a positive integer n suchthat q > 1/n. The definition of “greater than” ensures that x > 1/n. �

From time to time we shall revisit some of our earlier Brouwerian examples inthe light of our formal definitions of real numbers and their properties.

Proposition 2.1.7. The statement

∀x,y ∈ R (¬ (x � y)=⇒ y > x) (2.1)implies Markov’s principle.

Proof. Assume (2.1). Let (an)n�1 be an increasing binary sequence such that

¬∀n (an = 0) ,and define a real number by

x ={(

0,1n

): an = 0

}∪{(

1n

,1n

): an = 1 − an−1

}. (2.2)

Then ¬ (0 � x): for if 0 � q for all (q, q′) ∈ x, then an = 0 for all n, a contradiction.It follows from (2.1) that x > 0 and hence that there exists (q, q′) ∈ x such thatq > 0. Then (q, q′) =

( 1n ,

1n

)for a (unique) n such that an = 1 − an−1. �

Proposition 2.1.8. The relation > is cotransitive: If a < b, then for all x ∈ Reither a < x or x < b.

Proof. There exist (q, q′) ∈ a and (r, r′) ∈ b such that q′ < r. Given a real numberx, we can find (s, s′) ∈ x such that s′ − s < r − q′. If s′ < r, then x < b; if s′ � r,then q′ < s and so a < x. �

Lemma 2.1.9. If x > y � z or x � y > z, then x > z and x � z.

Proof. Assume, for example, that x > y � z. By Proposition 2.1.8, either x > z orz > y. Since the latter is ruled out by Lemma 2.1.4, we have x > z and therefore,by Lemma 2.1.5, x � z. �

Lemma 2.1.10. If (q, q′) ∈ x, then q � x � q′.

Proof. By Lemma 2.1.1, for each (r, r′) ∈ x, since the rational intervals [r, r′] and[q, q′] intersect, r′ � q and q′ � r. The desired conclusion now follows from thedefinition of the relation �. �

2.1 The Real Number Line 29

Lemma 2.1.11. For each real number x there exist rational numbers q,q′ suchthat q < x < q′.

Proof. Let (r, r′) be any element of x. By Lemma 2.1.10, r � x � r′. Choosingq, q′ in Q with q < r � r′ < q′, we see from the definition of the relation < thatq < x < q′. �

The following two propositions show that the classical law of trichotomy doesnot hold constructively, even in the weak form discussed in Proposition 2.1.13.


∀x ∈ R (x � 0 =⇒ x > 0 ∨ x = 0)

implies LPO.

Proof. Given an increasing binary sequence (an)n�1 , define the real number x asat (2.2). It is routine to check that x � 0; that if x = 0, then an = 0 for all n; andthat if x > 0, then there exists n such that an = 1. �


∀x ∈ R (x � 0 ∨ x � 0)

implies LLPO.

Proof. Let (an)n�1 be a binary sequence with at most one term equal to 1, anddefine a real number by

x ={(

− 1n

,1n

): ∀k � n (ak = 0)

}∪{(

(−1)n 1n

, (−1)n 1n

): an = 1

}.

If x � 0, then it is impossible that an = 1 for an odd n, so an = 0 for all odd n.Likewise, if x � 0, then an = 0 for all even n. �

Lemma 2.1.14. If x < y, then there exists s ∈ Q such that x < s < y.

Proof. Since x < y, there exist (q, q′) ∈ x and (r, r′) ∈ y such that q′ < r. Thenthe rational number

s ={(

12

(q′ + r) ,12

(q′ + r))}

has the desired property. �


The maximum of

Universitext - The Eye of Constructive... · Lume ce gˆandeaˆın basme ¸si vorbeaˆın poezii,...

Documents

Transcript of Universitext - The Eye of Constructive... · Lume ce gˆandeaˆın basme ¸si vorbeaˆın poezii,...